working on the logrel for soundness proof (#86)

* working on the logrel for soundness proof

* add end of section to make CI happy

* make CI happy

* CI is stubborn

* add natural numbers

* add the case for nat

* handled neutral case

* need to think of the right definition for gluing of pi

* finish gluing model

* add induction principle

* define wf gluing model

* add induction principle

* add extreme relations

* adjust the gluing model

theories/Core/Soundness/LogicalRelation.v

@@ -0,0 +1,269 @@
+From Coq Require Import Relation_Definitions RelationClasses.
+From Equations Require Import Equations.
+
+From Mcltt Require Import Base LibTactics.
+From Mcltt.Core Require Import System.Definitions Evaluation Readback PER.Definitions.
+From Mcltt Require Export Domain.
+From Mcltt.Core.Soundness Require Export Weakening.
+
+Import Domain_Notations.
+Global Open Scope predicate_scope.
+
+Generalizable All Variables.
+
+Notation "'typ_pred'" := (predicate (Tcons ctx (Tcons typ Tnil))).
+Notation "'glu_pred'" := (predicate (Tcons ctx (Tcons exp (Tcons typ (Tcons domain Tnil))))).
+
+Definition univ_typ_pred j i : typ_pred := fun Γ T => {{ Γ ⊢ T ≈ Type@j :  Type@i }}.
+Arguments univ_typ_pred j i Γ T/.
+
+Inductive glu_nat : ctx -> exp -> domain -> Prop :=
+| glu_nat_zero :
+  `( {{ Γ ⊢ m ≈ zero : ℕ }} ->
+     glu_nat Γ m d{{{ zero }}} )
+| glu_nat_succ :
+  `( {{ Γ ⊢ m ≈ succ m' : ℕ }} ->
+     glu_nat Γ m' a ->
+     glu_nat Γ m d{{{ succ a }}} )
+| glu_nat_neut :
+  `( per_bot c c ->
+     (forall {Δ σ v}, {{ Δ ⊢w σ : Γ }} -> {{ Rne c in length Δ ↘ v }} -> {{ Δ ⊢ m [ σ ] ≈ v : ℕ }}) ->
+     glu_nat Γ m d{{{ ⇑ ℕ c }}} ).
+
+Definition nat_typ_pred i : typ_pred := fun Γ M => {{ Γ ⊢ M ≈ ℕ : Type@i }}.
+Arguments nat_typ_pred i Γ M/.
+
+Definition nat_glu_pred i : glu_pred := fun Γ m M a => nat_typ_pred i Γ M /\ glu_nat Γ m a.
+Arguments nat_glu_pred i Γ m M a/.
+
+Definition neut_typ_pred i C : typ_pred :=
+  fun Γ M => {{ Γ ⊢ M : Type@i }} /\
+            (forall Δ σ V, {{ Δ ⊢w σ : Γ }} -> {{ Rne C in length Δ ↘ V }} -> {{ Δ ⊢ M [ σ ] ≈ V : Type@i }}).
+Arguments neut_typ_pred i C Γ M/.
+
+Inductive neut_glu_pred i C : glu_pred :=
+| ngp_make : forall Γ m M A c,
+    neut_typ_pred i C Γ M ->
+    per_bot c c ->
+    (forall Δ σ V v, {{ Δ ⊢w σ : Γ }} ->
+                {{ Rne C in length Δ ↘ V }} ->
+                {{ Rne c in length Δ ↘ v }} ->
+                {{ Δ ⊢ m [ σ ] ≈ v : M [ σ ] }}) ->
+    neut_glu_pred i C Γ m M d{{{ ⇑ A c }}}.
+
+Inductive pi_typ_pred i
+  (IR : relation domain)
+  (IP : typ_pred)
+  (IEl : glu_pred)
+  (OP : forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ IR }}),  typ_pred) : typ_pred :=
+| ptp_make : forall Γ IT OT M,
+    {{ Γ ⊢ M ≈ Π IT OT : Type@i }} ->
+    {{ Γ , IT ⊢ OT : Type@i }} ->
+    (forall Δ σ, {{ Δ ⊢w σ : Γ }} -> IP Δ {{{ IT [ σ ] }}}) ->
+    (forall Δ σ m a, {{ Δ ⊢w σ : Γ }} -> IEl Δ m {{{ IT [ σ ] }}} a -> forall (Ha : IR a a), OP _ _ Ha Δ {{{ OT [ σ ,, m ] }}}) ->
+    pi_typ_pred i IR IP IEl OP Γ M.
+
+Inductive pi_glu_pred i
+  (IR : relation domain)
+  (IP : typ_pred)
+  (IEl : glu_pred)
+  (elem_rel : relation domain)
+  (OEl : forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ IR }}),  glu_pred): glu_pred :=
+| pgp_make : forall Γ IT OT m M a,
+    {{ Γ ⊢ m : M }} ->
+    elem_rel a a ->
+    {{ Γ ⊢ M ≈ Π IT OT : Type@i }} ->
+    {{ Γ , IT ⊢ OT : Type@i }} ->
+    (forall Δ σ, {{ Δ ⊢w σ : Γ }} -> IP Δ {{{ IT [ σ ] }}}) ->
+    (forall Δ σ m' b, {{ Δ ⊢w σ : Γ }} -> IEl Δ m' {{{ IT [ σ ] }}} b -> forall (Ha : IR b b),
+                   exists ab, {{ $| a & b |↘ ab }} /\ OEl _ _ Ha Δ {{{ m [ σ ] m' }}} {{{ OT [ σ ,, m' ] }}} ab) ->
+    pi_glu_pred i IR IP IEl elem_rel OEl Γ m M a.
+
+
+Lemma pi_glu_pred_pi_typ_pred : forall i IR IP IEl (OP : forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ IR }}),  typ_pred) elem_rel OEl Γ m M a,
+    pi_glu_pred i IR IP IEl elem_rel OEl Γ m M a ->
+    (forall Δ m' M' b c c' (Hc : IR c c'), OEl _ _ Hc Δ m' M' b -> OP _ _ Hc Δ M') ->
+    pi_typ_pred i IR IP IEl OP Γ M.
+Proof.
+  inversion_clear 1; econstructor; eauto.
+  intros.
+  edestruct H5 as [? []]; eauto.
+Qed.
+
+Section Gluing.
+  Variable
+    (i : nat)
+      (glu_univ_typ_rec : forall {j}, j < i -> domain -> typ_pred).
+
+  Definition univ_glu_pred' {j} (lt_j_i : j < i) : glu_pred :=
+    fun Γ m M A =>
+      {{ Γ ⊢ m : M }} /\ {{ Γ ⊢ M ≈ Type@j : Type@i }} /\
+        per_univ j A A /\
+        glu_univ_typ_rec lt_j_i A Γ m.
+
+  #[global]
+    Arguments univ_glu_pred' {j} lt_j_i Γ m M A/.
+
+  Inductive glu_univ_elem_core : typ_pred -> glu_pred -> domain -> domain -> Prop :=
+  | glu_univ_elem_core_univ :
+    `{ forall typ_rel
+         el_rel
+         (lt_j_i : j < i),
+          j = j' ->
+          typ_rel <∙> univ_typ_pred j i ->
+          el_rel <∙> univ_glu_pred' lt_j_i ->
+          glu_univ_elem_core typ_rel el_rel d{{{ 𝕌@j }}} d{{{ 𝕌@j' }}} }
+
+  | glu_univ_elem_core_nat :
+    `{ forall typ_rel el_rel,
+          typ_rel <∙> nat_typ_pred i ->
+          el_rel <∙> nat_glu_pred i ->
+          glu_univ_elem_core typ_rel el_rel d{{{ ℕ }}} d{{{ ℕ }}} }
+
+  | glu_univ_elem_core_pi :
+    `{ forall (in_rel : relation domain)
+         IP IEl
+         (OP : forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), typ_pred)
+         (OEl : forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), glu_pred)
+         typ_rel el_rel
+         (elem_rel : relation domain),
+          glu_univ_elem_core IP IEl a a' ->
+          per_univ_elem i in_rel a a' ->
+          (forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}) b b',
+              {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+              {{ ⟦ B' ⟧ p' ↦ c' ↘ b' }} ->
+              glu_univ_elem_core (OP _ _ equiv_c_c') (OEl _ _ equiv_c_c') b b') ->
+          per_univ_elem i elem_rel d{{{ Π a p B }}} d{{{ Π a' p' B' }}} ->
+          typ_rel <∙> pi_typ_pred i in_rel IP IEl OP ->
+          el_rel <∙> pi_glu_pred i in_rel IP IEl elem_rel OEl ->
+          glu_univ_elem_core typ_rel el_rel d{{{ Π a p B }}} d{{{ Π a' p' B' }}} }
+
+  | glu_univ_elem_core_neut :
+    `{ forall typ_rel el_rel,
+          {{ Dom b ≈ b' ∈ per_bot }} ->
+          typ_rel <∙> neut_typ_pred i b ->
+          el_rel <∙> neut_glu_pred i b ->
+          glu_univ_elem_core typ_rel el_rel d{{{ ⇑ a b }}} d{{{ ⇑ a' b' }}} }.
+
+End Gluing.
+
+#[export]
+Hint Constructors glu_univ_elem_core : mcltt.
+
+
+Equations glu_univ_elem (i : nat) : typ_pred -> glu_pred -> domain -> domain -> Prop by wf i :=
+| i => glu_univ_elem_core i (fun j lt_j_i A Γ M => exists P El, glu_univ_elem j P El A A /\ P Γ M).
+
+
+Definition glu_univ (i : nat) (A : domain) : typ_pred :=
+  fun Γ M => exists P El, glu_univ_elem i P El A A /\ P Γ M.
+
+Definition univ_glu_pred j i : glu_pred :=
+    fun Γ m M A =>
+      {{ Γ ⊢ m : M }} /\ {{ Γ ⊢ M ≈ Type@j : Type@i }} /\
+        per_univ j A A /\
+        glu_univ j A Γ m.
+
+Section GluingInduction.
+
+  Hypothesis
+    (motive : nat -> typ_pred -> glu_pred -> domain -> domain -> Prop)
+
+      (case_univ :
+        forall i (j j' : nat)
+          (typ_rel : typ_pred) (el_rel : glu_pred) (lt_j_i : j < i),
+          j = j' ->
+          (forall P El A B, glu_univ_elem j P El A B -> motive j P El A B) ->
+          typ_rel <∙> univ_typ_pred j i ->
+          el_rel <∙> univ_glu_pred j i ->
+          motive i typ_rel el_rel d{{{ 𝕌 @ j }}} d{{{ 𝕌 @ j' }}})
+
+      (case_nat :
+        forall i (typ_rel : typ_pred) (el_rel : glu_pred),
+          typ_rel <∙> nat_typ_pred i ->
+          el_rel <∙> nat_glu_pred i ->
+          motive i typ_rel el_rel d{{{ ℕ }}} d{{{ ℕ }}})
+
+      (case_pi :
+        forall i (a a' : domain) (B : typ) (p : env) (B' : typ) (p' : env) (in_rel : relation domain) (IP : typ_pred)
+          (IEl : glu_pred) (OP : forall c c' : domain, {{ Dom c ≈ c' ∈ in_rel }} -> typ_pred)
+          (OEl : forall c c' : domain, {{ Dom c ≈ c' ∈ in_rel }} -> glu_pred) (typ_rel : typ_pred) (el_rel : glu_pred)
+          (elem_rel : relation domain),
+          glu_univ_elem i IP IEl a a' ->
+          motive i IP IEl a a' ->
+          per_univ_elem i in_rel a a' ->
+          (forall (c c' : domain) (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}) (b b' : domain),
+              {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+              {{ ⟦ B' ⟧ p' ↦ c' ↘ b' }} -> glu_univ_elem i (OP c c' equiv_c_c') (OEl c c' equiv_c_c') b b') ->
+          (forall (c c' : domain) (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}) (b b' : domain),
+              {{ ⟦ B ⟧ p ↦ c ↘ b }} -> {{ ⟦ B' ⟧ p' ↦ c' ↘ b' }} -> motive i (OP c c' equiv_c_c') (OEl c c' equiv_c_c') b b') ->
+          per_univ_elem i elem_rel d{{{ Π a p B }}} d{{{ Π a' p' B' }}} ->
+          typ_rel <∙> pi_typ_pred i in_rel IP IEl OP ->
+          el_rel <∙> pi_glu_pred i in_rel IP IEl elem_rel OEl ->
+          motive i typ_rel el_rel d{{{ Π a p B }}} d{{{ Π a' p' B' }}})
+
+      (case_neut :
+        forall i (b b' : domain_ne) (a a' : domain)
+          (typ_rel : typ_pred)
+          (el_rel : glu_pred),
+          {{ Dom b ≈ b' ∈ per_bot }} ->
+          typ_rel <∙> neut_typ_pred i b ->
+          el_rel <∙> neut_glu_pred i b ->
+          motive i typ_rel el_rel d{{{ ⇑ a b }}} d{{{ ⇑ a' b' }}})
+  .
+
+
+  #[local]
+    Ltac def_simp := simp glu_univ_elem in *.
+
+  #[derive(equations=no, eliminator=no), tactic="def_simp"]
+    Equations glu_univ_elem_ind i P El a b
+    (H : glu_univ_elem i P El a b) : motive i P El a b by wf i :=
+  | i, P, El, a, b, H =>
+      glu_univ_elem_core_ind
+        i
+        (fun j lt_j_i A Γ M => glu_univ j A Γ M)
+        (motive i)
+        (fun j j' typ_rel el_rel lt_j_i Heq Htr Hel =>
+           case_univ i j j' typ_rel el_rel lt_j_i
+             Heq
+             (fun P El A B H => glu_univ_elem_ind j P El A B H)
+             Htr
+             Hel)
+        (case_nat i)
+        _ (* (case_pi i) *)
+        (case_neut i)
+        P El a b
+        _.
+  Next Obligation.
+    eapply (case_pi i); def_simp; eauto.
+  Qed.
+
+End GluingInduction.
+
+
+Inductive glu_neut i Γ m M c A B : Prop :=
+| glu_neut_make : forall P El,
+    {{ Γ ⊢ m : M }} ->
+    glu_univ_elem i P El A B ->
+    per_bot c c ->
+    (forall Δ σ a, {{ Δ ⊢w σ : Γ }} -> {{ Rne c in length Δ ↘ a }} -> {{ Δ ⊢ m [ σ ] ≈  a : M [ σ ] }}) ->
+    glu_neut i Γ m M c A B.
+
+
+Inductive glu_normal i Γ m M a A B : Prop :=
+| glu_normal_make : forall P El,
+    {{ Γ ⊢ m : M }} ->
+    glu_univ_elem i P El A B ->
+    per_top d{{{ ⇓ A a }}} d{{{ ⇓ B a }}} ->
+    (forall Δ σ b, {{ Δ ⊢w σ : Γ }} -> {{ Rnf ⇓ A a in length Δ ↘ b }} -> {{ Δ ⊢ m [ σ ] ≈  b : M [ σ ] }}) ->
+    glu_normal i Γ m M a A B.
+
+
+Inductive glu_typ i Γ M A B : Prop :=
+| glu_typ_make : forall P El,
+    {{ Γ ⊢ M : Type@i }} ->
+    glu_univ_elem i P El A B ->
+    per_top_typ A B ->
+    (forall Δ σ a, {{ Δ ⊢w σ : Γ }} -> {{ Rtyp A in length Δ ↘ a }} -> {{ Δ ⊢ M [ σ ] ≈  a : Type@i }}) ->
+    glu_typ i Γ M A B.

theories/Core/Soundness/Weakening.v

@@ -0,0 +1 @@
+From Mcltt.Core.Soundness Require Export Weakening.Definition.

theories/Core/Soundness/Weakening/Definition.v

@@ -0,0 +1,19 @@
+From Mcltt Require Import Base System.Definitions.
+Import Syntax_Notations.
+
+Generalizable All Variables.
+
+Reserved Notation "Γ ⊢w σ : Δ" (in custom judg at level 80, Γ custom exp, σ custom exp, Δ custom exp).
+
+Inductive weakening : ctx -> sub -> ctx -> Prop :=
+| wk_id :
+  `( {{ Γ ⊢s σ ≈ Id : Δ }} ->
+     {{ Γ ⊢w σ : Δ }} )
+| wk_p :
+  `( {{ Γ ⊢w τ : Δ , T }} ->
+     {{ Γ ⊢s σ ≈ Wk ∘ τ : Δ }} ->
+     {{ Γ ⊢w σ : Δ }} )
+where "Γ ⊢w σ : Δ" := (weakening Γ σ Δ) (in custom judg) : type_scope.
+
+#[export]
+ Hint Constructors weakening : mcltt.

theories/LibTactics.v

@@ -1,4 +1,4 @@
-From Coq Require Export Program.Equality Program.Tactics Lia.
+From Coq Require Export Program.Equality Program.Tactics Lia RelationClasses.
 
 Create HintDb mcltt discriminated.
 
@@ -181,5 +181,19 @@ Ltac mautosolve := unshelve solve [mauto]; solve [constructor].
 #[export]
   Hint Extern 1 => eassumption : typeclass_instances.
 
+Ltac predicate_resolve :=
+  lazymatch goal with
+  | |- @Reflexive _ (@predicate_equivalence _) =>
+      simple apply @Equivalence_Reflexive
+  | |- @Symmetric _ (@predicate_equivalence _) =>
+      simple apply @Equivalence_Symmetric
+  | |- @Transitive _ (@predicate_equivalence _) =>
+      simple apply @Equivalence_Transitive
+  end.
+
+#[export]
+ Hint Extern 1 => predicate_resolve : typeclass_instances.
+
+
 (* intuition tactic default setting *)
 Ltac Tauto.intuition_solver ::= auto with mcltt core solve_subterm.

theories/_CoqProject

@@ -35,6 +35,9 @@
 ./Core/Syntactic/System.v
 ./Core/Syntactic/System/Definitions.v
 ./Core/Syntactic/System/Lemmas.v
+./Core/Soundness/LogicalRelation.v
+./Core/Soundness/Weakening.v
+./Core/Soundness/Weakening/Definition.v
 ./Frontend/Elaborator.v
 ./Frontend/Parser.v
 ./Frontend/ParserExtraction.v